Analogy Between Linear Optical Systems and Linear Two-Port Electrical Networks

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University of New Orleans ScholarWorks@UNO

Electrical Engineering Faculty Publications Department of Electrical Engineering

10-1972

Rasheed M.A. Azzam University of New Orleans, [email protected]

N. M. Bashara

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Recommended Citation R. M. A. Azzam and N. M. Bashara, "Analogy Between Linear Optical Systems and Linear Two-Port Electrical Networks," Appl. Opt. 11, 2210-2214 (1972) http://www.opticsinfobase.org/ao/ abstract.cfm?URI=ao-11-10-2210.

This Article is brought to you for free and open access by the Department of Electrical Engineering at ScholarWorks@UNO. It has been accepted for inclusion in Electrical Engineering Faculty Publications by an authorized administrator of ScholarWorks@UNO. For more information, please contact [email protected]. Analogy Between Linear Optical Systems and Linear Two-Port Electrical Networks

R. M. A. Azzam and N. M. Bashara

Attention is called to the analogy between linear optical systems and linear two-port electrical networks. For both, the transformation of a pair of oscillating quantities between input and output is of interest. The mapping of polarization by an optical system and of impedance (admittance) by a two-port network is described by a bilinear transformation. Therefore for each transfer property of a system of one type, there is a similar property for the system of the other type. Two-port electrical networks are synthesized whose impedance-(or admittance-) mapping properties are the same as the polarization-mapping properties of a given optical system. The opposite problem of finding the optical analogs of two-port networks is also considered. Besides unifying the methods of handling these two different kinds of systems, the analogy appears fruitful if used reciprocally to simulate electrical networks by optical systems, and vice versa. Linear mechanoacoustic systems have optical analogs besides their well-known electrical analogs.

1. Introduction outgoing waves are monochromatic and of the same The advent of microwaves and more recently of lasers frequency, and both are totally polarized. The polari- has stimulated the interest of electrical engineers in zation states of these waves can be specified with refer- optics and optical systems. Establishing analogies ence to any chosen pair of basis states, which are gener- between electrical networks and optical systems would ally elliptic. Any polarization state can be generated be useful in (1) unifying the methods of treating both by a linear superposition of the basis states 1 and 2 using kinds of systems and (2) the reciprocal simulation of complex coefficients E and E2, respectively. These systems of one type by systems of the other. coefficients transform linearly after propagation through the system S

11. Basis of the Analogy /El'>\ T., T12 (El\ (1) The basis of the analogy between optical systems and \E2'/ ( T2, T22 E2 two-port electrical networks' is explained with reference where Tij denote the complex elements of the generalized to Fig. 1. For the optical system S, shown in Fig. 1 (a), Jones matrix2 of the optical system. The matrix T nonlinear optical effects and other frequency-changing depends both on the internal structure of S and on the phenomena (such as Brillouin or Raman scattering) are basis states used at the input and output of the optical assumed absent. Also, incoherent scattering processes system, which need not be the same. When two orthog- are excluded. Under these conditions the incident and onal linear polarizations are chosen as basis states, Eq. (1) becomes

W vi (2) E Y \T21 T22IEI where (T,") refers to the ordinary Cartesian Jones ma- (a) (b) trix and the elements of the vector E correspond to the Fig. 1. (a) Linear optical system S; (b) linear two-port network Cartesian components of the electric vector of the light W. For S and W the transformation of a pair of oscillating wave. quantities between input and output is of interest. In many cases we are interested in the polarization- mapping properties of the system S overlooking any over-all amplitude or phase changes between input and, The authors are with the Electrical Materials Laboratory, output. The polarization forms (or the ellipses of Department of Electrical Engineering, University of Nebraska, polarization) of the incident and outgoing waves are Lincoln, Nebraska 68508. determined by the complex ratios X and x', respectively, Received 13 March 1972. where

2210 APPLIED OPTICS / Vol. 11, No. 10 / October 1972 x = E2/E,, x = E2//Ell' (3) (?-- - f72) = ( 3 - - 2)/(l3 - 772) (3 - 1)] According to Eq. (1) x' and x are interrelated by X [(¢ - 1)/(¢- 2)1- (9)

x' = (T22x + T21)/(Tl2x + T1), (4) This means that if the transformation of three polarization states by an optical system S or three impedances which is a bilinear transformation. The result in Eq. by a network W is known, the mapping of all other (4) has been utilized'- 6 to investigate various aspects of polarizations or impedances is completely determined. polarization transfer by optical systems and to provide The bilinear transformation is referred to as the polari- a procedure' (generalized ellipsometry) to measure the zation transfer function (PTF) or the impedance matrix T. transfer function (ITF). According to Eqs. (4) ahd (7), The two-port network shown in Fig. 1(b) is assumed the coefficients of the PTF and the ITF determine the linear and may contain active elements operating in the matrices T and W of Eqs. (1) and (5), respectively, up to linear domain.7 The voltage and current V' and I' at a complex multiplier. Besides the mapping of three one port are related to the voltage and current V and I polarizations or three impedances, a fourth measure- at the other port by a linear transformation ment is needed to determine these matrices completely. ( Wv1 12 \ I \ This involves the determination of the over-all phase (I) = delay and attenuation of a light wave of a given polari- W21 W22 V} zation by the optical system and a complex voltage- or where Wfj are the complex elements of the network 8 current-transfer ratio for the network. transfer matrix. The impedances at the two ports For any choice of the basis states Eq. (3) leads to a are defined by complex-plane representation in which the loci of elliptic = V/I, Z = V'/I', (6) polarizations of the same azimuth and of the same ellipticity constitute two orthogonal sets of circles. These and according to Eq. (5) they are related by equiazimuth and equiellipticity families of circles re-

Z = (W Z + W )/(Wl Z + W11), (7) spectively pass through and enclose the two points in 2 2 21 2 the complex plane that represent the right and left which is a bilinear transformation. Obviously, the circular polarizations. If the latter polarizations are transformation of admittance between the two ports is chosen as the basis states, the equiazimuth and equi- also bilinear, ellipticity contours become straight lines and concentric circles through and around the origin, respectively. In Y' = (W,1 Y + W12)/(W21Y + W22). (8) the impedance plane these straight lines and circles The analogy between optical systems and two-port through and around the origin correspond to the loci of networks is clearly demonstrated by the similarity of impedances of constant angle and of constant magnitude, Eqs. (1) and (5) and Eqs. (4) and (7) or (8). In both respectively. The origin and the point at infinity cor- cases we deal with the transformation of (the complex respond to the left and right circular polarizations in one amplitudes of) a pair of oscillating quantities between case and to zero impedance (short circuit) and infinite the input and output. These are the two oscillating impedance (open circuit) in the other. components of the light vector along the two chosen An important property of bilinear transformations basis states in the case of optical systems and the voltage between two complex planes is that a circle in one plane and current oscillations in the case of two-port networks. is mapped into a circle in the other plane. Applying Rumsey has previously shown the similarity of the this property, it is seen that if an incident polarization definition of impedance and polarization ratios and x is of a fixed azimuth (ellipticity), the polarization x' suggested the use of impedance charts (Smith or Carter) exiting from the optical system S will traverse a -circle 9 to represent elliptical polarizations. Deschamps briefly in the complex plane as the ellipticity (azimuth) of x is mentioned the similarity between the mapping of varied to scan all possible values.' Similarly, for the polarization by optical systems and of impedance by two-port network, W, if an impedance Z of constant 0 electrical networks. ' angle (magnitude) is connected at one port of the inpe- 111. Analogous Properties of Optical Systems dance, Z' that appears at the other port will follow a circle in the complex plane as the magnitude (angle) of and Two-Port Networks Z is varied to cover all possible values. The optical From the previous section we have seen that the system S and the two-port network W map the two mapping of polarization by an optical system and of orthogonal families of straight lines and concentric impedance (or admittance) by a two-port electrical circles through and around the origin in the polariza- network between input and output is described by a tion-impedance (-Z) plane into two orthogonal sets of bilinear transformation. It follows that for each ter- circles in the ('-Z') plane (Fig. 2). The preservation minal characteristic of a system of one type there is a of orthogonality is a consequence of the conformal na- similar characteristic for the system of the other type. ture of the transformation. The two points A' and B' First we note that a bilinear transformation can al- in the ('-Z') plane are the images of the origin A and ways be found that maps any three points (, 2, 3) in the point at infinity B of the (-Z) plane. A' and B' the complex v plane into the three points (7j,, 2, 3) in represent the response of S to incident left and right the complex 77plane and is given by3"1 circular polarizations and they also represent the impe-

October 1972 / Vol. 11, No. 10 / APPLIED OPTICS 2211 In!1= 1r, (13) or ((A- + B)/(CP + D) = If. (14) Similarly the locus of polarizations that preserve their azimuth or impedances that preserve their angle is given by 2 Arg(,7) = Arg(), (15) or (C-Z) Plane ( Z-)Plane Arg[(At + B)/(CD + D)l = Arg(). (16) Fig. 2. The two orthogonal families of straight lines (-) and concentric circles (-) through and around the origin in the Expanding Eqs. (14) and (16) yields 2 polarization-impedance (x-Z) plane (left) represent polarization (x + y)Ql - Q2 = 0, (17) states of equiazimuth and equiellipticity and impedances of constant angle and constant magnitude, respectively. These and are mapped by the optical system S and the two-port netowrk W xQ - yQ = 0, (18) into the orthogonal families of circles passing through and en- 3 closing the two points A' and B' (right). A' and B' in the (x'- respectively. In Eqs. (17) and (18) we have Z') plane are the images of the origin (A) and the point at infinity (B) of the (x-Z) plane. = x + jy, and the Q's are quadratics in x and y given by Q = (C-C)(x' + y) + (C-D)x + 2(C X D)y + (D-D), dance that appears at one part of W when the other port Q2 = (A-A)(x3 + y) + 2(A-B)x + 2(A X B)y + (B-B), is short- and open-circuited, respectively. Q3 = (C X A)(X2 + y)- (B X C - D X A)x For each optical system there are two polarization states that pass through the system unchanged. These - (B-C - D-A)y + (D X B), (20) 2 eigenpolarizations' are obtained from Eq. (4) by setting Q4 = (C-A)(X + y2) + (B-C + D-A)x x' = x. This yields - (B X C + D X A)y + (D*B). Xd,2 = (1/27'12) In Eq. (20) A, B, C, and D are the complex coefficients 2 X {(T22 -2',) F(T22-Ti) + 4T,2T2,,ll. (10) of the PTF or the ITF considered as vectors in the com- Corresponding to the eigenpolarizations Xel and Xe2 of plex plane. The operations of the dot and cross prod- the optical system S, the two-port network W has two ucts have their usual meanings except that for the iterative impedances Zil and Z1, that when connected at cross product only the magnitude (with proper sign) is one port appear unchanged at the other. By analogy, to be taken. Equations (17)-(20) can be easily used to Eq. (7) gives determine these loci for any optical system or two-port network. The explicit form of the equations of the loci Zil,2 = (1/2W12) of invariant-ellipticity states and invariant-azimuth X (W22 - W1) A: [(W22 - W1,)2 + 4W12W 21 }. (11) states of an optical system in the cartesian complex- plane representation can be found in Ref. 6. Because Eqs. (10) and (11) involve ratios of the elements of the transfer matrices T and W, the eigenpolarizations and iterative impedances are uniquely determined by IV. Simulation of Optical Systems by the PTF and ITF, respectively. Equivalent Two-Port Networks and Vice Versa The loci of polarization states that preserve either Because of the analogy between the terminal char- ellipticity or azimuth after propagation through an acteristics-1of optical systems and two-port networks, it optical system6 are analogous to the loci of impedances is possible to simulate systems of one type by systems of that if connected at one port would appear at the other the other. The equivalence is achieved when the PTF with either magnitude or angle unchanged, respectively. of the optical system [Eq. (4)] and the ITF (or the To determine the cartesian equation of these loci, both admittance transfer function) of the two-port network Eqs. (4) and (7) are put in the form [Eq. (7) or (8) ] are identical. For a given optical system (two-port network) there - = (AD + B)/(C- + D), (12) is one and only one PTF (ITF). However, the opposite where v = x or Z, - = x' or Z', and the coefficients A, is not true, namely, that there can be many optical B, C, and D correspond to the elements of either the T systems (two-port networks), all of which have the matrix of the optical system or the W matrix of the same PTF (ITF). In other words, there can be many two-port network. The locus of polarizations that possibilities for the internal structure of S or W, all of preserve their ellipticity or impedances that preserve which lead to the same terminal polarization- or impe- their magnitude is determined by'2 dance-mapping properties.

2212 APPLIED OPTICS / Vol. 11, No. 10 / October 1972 Consider the following problem. Given an optical tion of a transmission line of characteristic impedance system S, find a two-port network W whose impedance- Z2 between them. A simpler (and more practical) or admittance-transforming properties are the same as synthesis is the T network of Fig. 3(b) whose impe- the polarization-mapping properties of S. The first dances ZI, Z2, and Z3 are given by step in the solution is to find the PTF of S. As is evi- Z = (T1/T1) - Z , Z = (det T)i/T12, dent from Eq. (4), the Jones matrix T has to be deter- 2 2 2 mined apart from a complex multiplying factor. Be- Z3- (T22/T12) - Z2- (27) cause T depends on the internal structure of S as well as The corresponding alternative solutions based on the the basis polarization states, the latter have to be chosen. mapping of admittance instead of impedance between Such a choice is determined by the system under con- the two ports according to Eq. (23) are the duals of sideration, and in most cases it is between a pair of or- in Fig. 3. Other network configurations circular those shown thogonal linear polarizations, the right and left with three independent impedances or admittances can polarizations or a mixture of these. Once the basis be found that satisfy either Eq. (22) or Eq. (23). states have been decided upon, the Jones matrix T is The opposite problem is to find the optical analog S obtained from of a given two-port network W. A direct procedure T = TN... T2T,, (21) would be to search for an optical system S with six real independent parameters (e.g., two dichroic retarders and . . ., indi- where T1, T2 and TN are the matrices of the a rotater) whose PTF is the same as the ITF or the ad- vidual devices encountered by the light beam, with T, mittance transfer function of W. In general the syn- referring to the first-to-be-encountered optical element. thesis is not as straightforward as in the first case, when Substituting the elements of the matrix T of Eq. (21) S is given and W is to be synthesized. into Eq. (4) the PTF of the optical system is found. If W , W , . . . , W, represent the two-port-network The ITF or the admittance transfer function of an 1 2 analogs of the optical systems S,, S2, . . S,,, then the equivalent two-port network W is obtained from Eq. network formed by connecting W1 , W2, . . ., and W,, (4) simply by substituting Z or Y for x and Z' or Y' for in succession will correspond to the cascade of the opti- x'. This gives cal systems S, S2, . . ., and S,, placed one after the Z' = (T22Z + T21)/(T12Z + Tji), (22) other along the direction of propagation and in the same order in which the networks are connected. or It is interesting to find the electrical analogs of some Y' = (T22Y + T21)/(Tl2Y + T,,). (23) of the simple optical elements. A polarizer is by definition an optical device that transforms any of the various Thus the equivalence is between the polarization-map- an incident light wave into a or the polarization forms of ping properties of S and either the impedance- unique polarization state at its output. In this case the admittance-mapping properties of W. To synthesize determinant T22T, - T21 T12 of the bilinear transforma- a two-port network that maps an impedance Z con- tion vanishes and Eq. (4) becomes nected at one port into an impedance Z' at the other port in accordance with Eq. (22) we can rewrite the x' = xo = T22/T 1 2 = '21/Til, (28) latter equation in the form where xo represents the outgoing polarization state, Z = Z3 + [Z22/(Z + Zl)], (24) which is generally elliptic. The equivalent circuit where analog is shown in Fig. 4(a), where Zo = xo. A polarizer is said to exhibit leakage if it has a small, nonzero trans- 1 Z = T,/T,,, Z2 = (det T) /T12 , Z3 = T22/T12, (25) mission for the incident polarization state that is ortho- and det T denotes the determinant of the bilinear trans- gonal with xo. This condition is taken care of by replac- formation, ing the short circuit of Fig. 4(a) by a small impedance Z, such that IZIf << ZoI [Fig. 4(b)]. As another ex- det T = TT22 - T12 T21. (26) ample, a linear dichroic plate shows nonequal but in- Equation (24) leads to the network of Fig. 3(a), which phase transmittances along two orthogonal principal shows two series impedances Z, and Z with a X/4 sec- directions in its plane. In this case it can be easily 3 shown that Eq. (4) degenerates to the simple linear transformation x' = Kx, (29) ZZ Z'> VZ2~~ UZ where K is the ratio of the amplitude transmittances along the two principal axes of dichroism. Obviously the circuit analog is an ideal transformer [Fig. 4(c)1 (a) (b) whose turns ratio is KI. A rotator is a device that Fig. 3. Two-port electric networks with impedance-mapping rotates the major axis of the polarization ellipse through properties identical to the polarization-mapping properties of a a fixed angle in a fixed sense for all orientations and given optical system. The parameters of the networks (a) and ellipticities of the incident polarization ellipse. The (b) are given by Eqs. (25) and (27), respectively. cartesian Jones matrix is

October 1972 / Vol. 11, No. 10 / APPLIED OPTICS 2213 zo zo port network is described by a bilinear transformation. For each terminal characteristic of a system of one type Z'ZOf nl~~z there is a similar characteristic for the system of the other type. For example, the two eigenpolarizations of the optical system are the analogs of the two iterative (a) (b) impedances of the two-port network. The loci of polarization states that preserve ellipticity or azimuth R3 R] and those of impedances (or admittance) that preserve magnitude or angle are described by the same equations Id) qZ in the complex plane. Two-port electrical networks are synthesized whose impedance- (or admittance-) mapping Ic) {d) properties are the same as the polarization-mapping properties of a given optical system. The opposite optical de- Fig. 4. The circuit analogs of some of the simple problem is also considered. The analogy presented in vices: (a) ideal polarizer, (b) imperfect polarizer with leakage 1Z2 << Z,, (c) purely dichroic plate, and (d) optical rotator. this paper is useful because (1) it unifies the methods of treating both kinds of systems and (2) it leads to the reciprocal simulation of systems of one type by systems of the other, which is of practical value. ( cosa sin 7'r = (30) This work was supported by the National Science -sina cosa Foundation. References which when substituted in Eq. (4) gives 1. When there is more than one incident beam or when a single incident beam generates a number of emergent beams (e.g., X = (X cosa - sina)/(X sina + cosa). (31) in the presence of a diffraction grating), the electrical analog of the optical system becomes a multiport network. There is no loss of generality, however, when a two-port network The parameters of the equivalent T network are is used to represent the relation between one incident and one emergent beam.

Z = - tan(a/2), Z2 = CSCa, Z3 = - tan(a/2), (32) 2. W. A. Shurcliff, Polarized Light (Harvard U.P., Cambridge, 1962). which follows from Eqs. (27) and (30), and leads to the 3. R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, resistive circuit of Fig. 4(d). Note that in this case the 222 (1972). equivalent circuit analog is not as simple as those en- 4. R. M. A. Azzam and N. M. Bashara, Optics Commun. 4, countered in the above two examples. 203 (1971). The simplicity of producing and measuring light of 5. R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, known polarization coupled with the fact that the optical 336 (1972); Opt. Commun. 5, 5 (1972). analogs of negative impedence networks are readily realizable may make the optical simulation of elec- 6. R. M. A. Azzam and N. M. Bashara, Appl. Opt. 12, January trical two-port networks of considerable practical value. (1973). However, the opposite does not appear to be as simple, 7. Large-signal equivalent circuits are sometimes used to relate namely, to simulate optical systems by electrical two- voltages and currents of the same frequency in the presence of port networks as can be seen from the example of the nonlinear effects [see, e.g., M.A.H. El-Said, IEEE Trans. optical rotator. Circuit Theory 17, 8 (1970)]. Such circuits might simulate Finally, it should be noted that linear mechanoacou- nonlinear optical systems if light waves of the same frequency stic systems are described by equations similar to Eqs. are coherent at input and output. (1) and (5). In this case the pair of oscillating quanti- 8. Conventionally voltages are listed before currents in the in- ties are the mechanical force (or pressure) and velocity put and output vectors. (or displacement). Therefore these systems have their 9. V. H. Rumsey, Proc. IRE 39, 535 (1951). optical analogs as well as their well-known electrical analogs. 10. G. A. Deschamps in Proc. Symp. Modern Network Synthesis J. Fox, Ed. (Polytechnic Press, Brooklyn, 1952). 11. E. A. Guillemin, Mathematics of Circuit Analysis, (Mas- V. Conclusions sachusetts Institute of Technology Press, Cambridge, 1949). 12. When the orthogonal left and right circular polarizations Analogy between optical systems and two-port elec- are used as basis states, the azimuth 0 and ellipticity e of the trical networks is demonstrated. In each case we deal ellipse of polarization are obtained from the complex polari- with the transformation of a pair of oscillating quanti- zation variable x [Eq. (3)] by the relations 0 = 1 Arg(x) ties between input and output. The mapping of polari- and = ( - 1x|i)/(IXl + 1). These lead to the equiazi- zation by an optical system and of impedance by a two- muth-equiellipticity chart of Fig. 2 (left).

2214 APPLIED OPTICS / Vol. 11, No. 10 / October 1972